The application of the wavelet transform to speech recognition, image processing, turbulence theory, non-linear, non-equilibrium physics, chaos and fractals has become extensive in recent years.
Wavelet analysis is signal processing that is used when it is desired to know not only the frequency contained in data but also time or position information. Wavelet analysis involves selecting, as a base function, a function localized about a certain time (space) and not a function spread uniformly in time (space), as in the manner of a trigonometric function used in Fourier analysis, and performing analysis of a signal by a parallel translation and similar transformation. By virtue of such analysis, it becomes possible to specify not only frequency but also the position of an event.
Further, since the signal is decomposed by a similar transformation, which is dilation and contraction of the base, the nature of wavelet analysis is such that time-wise resolution rises with respect to sudden changes (high frequencies) while frequency resolution rises with respect to gradual changes (low frequencies).
A wavelet transform T.sub.g (a,b) of an input signal S(t) in which G(t) represents the base function is defined as follows: ##EQU1## where * denotes the complex conjugate and T.sub.g (a,b) is referred to as a wavelet coefficient. Further, a represents a scale parameter [the range of dilation of the base function G(t)] and b represents shift (the position of the base function in space or time). Equation (1) is the convolution of the base function G(t) and input signal S(t).
By taking the discrete versions of the base function and input signal and representing these by G[(n-b)/a] and S(n) (n=1, 2, . . . ), respectively, we may express Equation (1) as follows: ##EQU2##
FIGS. 24a and 24b illustrate the distinction between a wavelet transform and a Fourier transform.
FIG. 24a shows the wavelet transform. Here the input signal S(t) is decomposed into signals G.sub..alpha.0,.omega.0 (t), G.sub..alpha.1,.omega.1 (t) localized about a certain time (space). Here 2.sub..omega.0 =.omega..sub.1 holds. The function G.sub..alpha.,.omega. (t) shown in FIG. 24a is the Gabor function, which is used generally as the base function, and is expressed by the following equation: ##EQU3##
FIG. 24b shows the Fourier transform. Here the input signal S(t) is decomposed into sine waves E.sub.1 sin (w.sub.f0t +.psi..sub.1), E.sub.2 sin (w.sub.f1 t+.psi..sub.2) dilated uniformly in time (space).
The Fourier transform generally has come to be applied widely owing to the development of an outstanding high-speed algorithm referred to as an "FFT" (Fast-Fourier Transform).
An algorithm for implementing high-speed processing in highly precise fashion has not yet been developed for the wavelet transform.
The problem is that when the wavelet coefficient is obtained in accordance with Equation (2), the support of the base function G used takes on a large size at low frequencies. As a consequence, the number of samplings increases even if the sampling interval is the same.
It has been proposed to reduce the number of samples by performing linear interpolation with regard to the base function when a wavelet coefficient of low frequency is obtained. Consider a situation in which the j-th wavelet coefficient is obtained. Assume that the base function is composed of N-number of sampling points G.sub.i (i=1, 2, . . . N) in the case of the (j-1)th base function. The j-th base function is constructed from these points by performing linear interpolation. There are instances in which incorrect results are obtained by adopting this method of interpolation.